Predicate Logic : Satisfiability

Question

I have a set as : {∀x ∃y P(x, y), ∀x¬P(x, x)}. . In order to satisfy this set I know that there should exist an interpretation I such that it should satisfy all the elements in the set. For instance my interpretation for x is 3 and for y is 4. Should I apply the same numbers (3,4) to ∀x¬P(x, x) as well ? Moreover, there are two x's in the function argument so should I apply 3 and 3 which is the value of the x that I assigned ? Thanks.

Answer 1

Take $\Omega$ any set containing at least two elements, and $P(x,y):x\ne y$. You have one interpretation :-)

Answer 2

Giving an interpretation means first of all specifying a set $I$ (the "universe") over which the variables range. In your case, you could choose $I = \{3,4\}$. Next, you have to choose how to interpret all the predicate symbols. Predicate symbols (i.e., syntactical objects) must be interpreted by relations over $I$ (i.e., semantical objects). For instance, you might interpret $P$ by the relation $P_I = \{(3,4),(4,3)\} \subseteq I^2$. And finally, you have to verify that all the formulas hold under this interpretation. In fact, they do: The first one holds, because for every $x \in I$ there exists some $y \in I$ such that $(x,y)$ is contained in $P_I$ (if $x = 3$, choose $y = 4$, if $x = 4$, choose $y = 3$). The second one holds as well, because for every $x \in I$, the pair $(x,x)$ is not contained in $P_I$.